I will describe how to use topological Azumaya algebras,
or, equivalently, principal ${PU_n}$-bundles, to think about two problems
in algebraic geometry: the period-index problem about the degrees
of division algebras over function fields, and the problem of the
existence of projective maximal orders in unramified division algebras.
In particular, using topological methods, I will show that projective
maximal orders do not necessarily exist, which solves an old problem
of Auslander and Goldman.

Brosnan, Patrick
University of Maryland

Algebraic groups related to Hodge theory

In Hodge theory, there are several categories of objects
that turn out to be (neutral) Tanakian, for example, split Hodge structures,
mixed Hodge structures, variations of Hodge structure, etc. As such
these categories are equivalent to the category of representations
of their Tanakian galois groups. Unfortunately, most of these groups
seem difficult to describe explicitly. However, there is an easy description
of the category of split real Hodge structures. It is the category
of representations of group Deligne called S: the Weil restriction
of scalars from C to R of the multiplicative group. Deligne also described
real mixed Hodge structures. But here the group involved is more complicated:
it is the semi-direct product of S with a pro-unipotent group scheme
U. Nilpotent orbits are certain variations of Hodge structure, which
can be defined in terms of linear algerbaic data. The simplest of
these are the SL2 orbits introduced by Schmid. It turns out that the
category of SL2 orbits is equivalent to the category of representations
of a certain real reductive algebraic group over R which is a semi-direct
product of SL2 and Deligne's group S. I will describe this and a related
group which controls certain nilpotent orbits. This gives a group-theoretic
understanding of certain operations on variations of mixed Hodge structure,
such as, taking the limit mixed Hodge structure.
The content of this talk is joint work with Gregory Pearlstein.

Brussel, Eric
California State Polytechnic University

Arithmetic in the Brauer group of the
function field of a p-adic curve

Joint work with: Kelly McKinnie and Eduardo Tengan. We present
machinery that allows us to prove several results concerning the
n-torsion subgroup of the Brauer group of the function field F of
a p-adic curve, when n is prime to p. We prove that every class
of period n is expressible as a sum of two Z/n-cyclic classes, and
a more general statement relating symbol lengths of function fields
of curves over a complete discretely valued field K and function
fields of curves over the residue field of K. We also reprove Saltman's
theorem that every division algebra of degree n (not p) over the
function field of a p-adic curve is cyclic.

Fedorov, Roman
Kansas State University

1. A conjecture of Grothendieck and Serre
on principal bundles

Let R be a regular local ring, G be a reductive R-group scheme.
A conjecture of Grothendieck and Serre predicts that a principal
G-bundle on spec(R) is trivial if it has a rational section.This
has been proved in many particular cases. Recently Fedorov and Panin,
using previous results of Panin, Stavrova and Vavilov, gave a proof
in the case, when R contains an infinite field.

I will discuss the statement of the conjecture, some corollaries,
and the strategy of the proof.

2. A conjecture of Grothendieck and Serre and affine Grassmannians

This is a continuation of my previous talk. I will introduce affine
Grassmannians parameterizing modifications of principal G-bundles
on the projective line over a scheme. While the proof discussed
in my first talk does not formally rely on affine Grassmannians,
they were crucial in creating this proof.

Then I will explain how one can use affine Grassmannians to construct
certain "exotic" principal bundles. This will explain
why certain "naive" attempts at a proof of Grothendieck-Serre
conjecture failed.

Gille, Stefan
University of Alberta

Permutation modules and motives

We discuss how permutation resolutions of Chow groupscan be used
to compute geometrically split motives (in some cases ).

Kuttler, Jochen
University of Alberta

Tensors of bounded ranks are defined in
bounded degree

Tensor rank is a very classical notion, naturally arising in algebraic
geometry, algebraic statistics, and complexity theory. In this context
an old problem is to determine the rank of a given tensor, that
is, to find defining equations for the variety of tensors of a given
(border) rank k. In this talk I will report on joint work with Jan
Draisma, where we prove qualitative results on the variety of p-tensors
of border rank at most k.
For example we show that this variety is defined by equations of
degree at most d = d(k), independent of the number of tensor factors
(or the dimension of each factor).

Lee, Ting-Yu
Fields Institute

The local-global principle for embeddings of maximal tori into
reductive groups

Let G be a reductive group, T be a torus and ${\Psi}$ be
a root datum associated with T. In this talk, I will discuss when
we can embed T to G as a maximal torus with respect to the root datum
${\Psi}$. Over local fields, the existence of such embedding is determined
by the Tits indices of G and ${\Psi}$. Then I will use this to construct
an example where the local-global principle for the embedding fails.
I will also explain the relation between the embeddings of root data
into reductive groups and embeddings of étale algebras with
involution into central simple algebras with involution. The latter
was discussed in G. Prasad and A. Rapinchuk's paper.

Lemire, Nicole
University of Western Ontario

Equivariant Birational Aspects of Algebraic
Tori

We examine the equivariant birational linearisation problem for
algebraic tori equipped with a finite group action. We also study
bounds on the degree of linearisability, a measure of the obstruction
for such an algebraic torus to be linearisable. We connect these
problems to the question of determining when an algebraic group
is (stably) Cayley - that is (stably) equivariantly birationally
isomorphic to its Lie algebra.
We discuss joint work with Popov and Reichstein on the classification
of the simple Algebraic groups which are Cayley and on determining
bounds on the Cayley degree of an algebraic group, a measure of
the obstruction for an algebraic group to be Cayley.
We also relate this to recent work with Borovoi, Kunyavskii and
Reichstein extending the classification of stably Cayley simple
groups from the algebraically closed characteristic zero case to
arbitrary fields of characteristic zero. Lastly, we investigate
the stable rationality of four-dimensional algebraic tori and the
associated equivariant birational linearisation problem.

Matzri, Eli
University of Virginia

Symbol length over C_r fields

A field, F, is called C_r if every homogenous form of degree n
in more then n^r variables has a non-trivial solution.
Consider a central simple algebra, A, of exponent n over a field
F. By the Merkurjev-Suslin theorem assuming F contains a primitive
n-th root of one, A is similar to the product of symbol algebras,
the smallest number of symbols required is called the length of
A denoted l(A).
If F is C_r we prove l(A) \leq n^{r-}-1. In particular the length
is independent of the index of A.

Merkurjev, Alexander
University of California, Los Angeles

On cohomological invariants
of semisimple groups

Neftin, Danny
University of Michigan, Ann Arbor

Noncrossed products over Henselian fields
and a Grunwald-Wang problem

A finite dimensional division algebra is called a crossed product
if it contains a maximal subfield which is Galois over its center,
otherwise a noncrossed product.

Since Amitsur settled the long standing open problem of existence
of noncrossed products, their existence over familiar fields was
an object of investigation. The simplest fields over which they
occur are Henselian fields with global residue field (such as Q((x)),
where Q is the field of rational numbers). We shall describe the
"location" of noncrossed products over such fields by
proving the existence of bounds that, roughly speaking, separate
crossed and noncrossed products. Furthermore, we describe those
bounds in terms of Grunwald-Wang type of problems and address their
solvability in various cases.
(joint work with Timo Hanke and Jack Sonn)

Rapinchuk, Igor
Yale University

On the conjecture of Borel and Tits for
abstract homomorphisms of algebraic groups

The conjecture of Borel-Tits (1973) states that if $G$ and $G'$
are algebraic groups defined over infinite fields $k$ and $k'$,
respectively, with $G$ semisimple and simply connected, then given
any abstract representation $\rho \colon G(k) \to G' (k')$ with
Zariski-dense image, there exists a commutative finite-dimensional
$k'$-algebra $B$ and a ring homomorphism $f \colon k \to B$ such
that $\rho$ can essentially be written as a composition $\sigma
\circ F$, where $F \colon G(k) \to G(B)$ is the homomorphism induced
by $f$ and $\sigma \colon G(B) \to G'(k')$ is a morphism of algebraic
groups. We prove this conjecture in the case that $G$ is either
a universal Chevalley group of rank $\geq 2$ or the group $\mathbf{SL}_{n,
D}$, where $D$ is a finite-dimensional central division algebra
over a field of characteristic 0 and $n \geq 3$, and $k'$ is an
algebraically closed field of characteristic 0. In fact, we show,
more generally, that if $R$ is a
commutative ring and $G$ is a universal Chevalley-Demazure group
scheme of rank $ \geq 2$, then abstract representations over algebraically
closed fields of characteristic 0 of the elementary subgroup $E(R)
\subset G(R)$ have the expected description. We also describe some
applications of these results to character varieties of finitely
generated groups.

Rapinchuk, Andrei
University of Virginia

The genus of a division algebra and ramification

Let $D$ be a finite-dimensional central division algebra over a
field $K$. The genus $\mathbf{gen}(D)$ is defined to be the set
of the Brauer classes $[D'] \in \mathrm{Br}(K)$ where $D'$ is a
central division $K$-algebra having the same maximal subfields as
$D$. I will discuss the ideas involved in the proof of the following
finiteness result: {\it Let $K$ be a finitely generated field, $n
\geqslant 1$ be an integer prime to $\mathrm{char} \: K$. Then for
any central division $K$-algebra $D$ of degree $n$, the genus $\mathbf{gen}(D)$
is finite.} One of the main ingredients is the analysis of ramification
at a suitable chosen set of discrete valuations of $K$. Time permitting,
I will discuss generalizations of these methods to absolutely almost
simple algebraic groups. This is a joint work with V.~Chernousov
and I.~Rapinchuk.

Saltman, David
Princeton University

Finite u Invariants and Bounds on Cohomology
Symbol Lengths

We answer a question of Parimala's showing that fields with finite
u invariant have bounds on the symbol lengths in their $\mu_2$ cohomology
in all degrees.

Stavrova, Anastasia
Fields Institute

On the congruence kernel of isotropic groups
over rings

We discuss an extension of a recent result of A. Rapinchuk and
I. Rapinchuk on the centrality of the congruence kernel of the elementary
subgroup of a Chevalley (i.e. split) simple algebraic group to the
case of isotropic groups. Namely, we prove that for any simply connected
simple group scheme G of isotropic rank at least 2 over a Noetherian
commutative ring R, the congruence kernel of its elementary subgroup
E(R) is central in E(R). Along the way, we define the Steinberg
group functor St(-) associated to an isotropic group G as above,
and show that for a local ring R, St(R) is a central covering of
E(R).

Vavilov, Nikolai
St. Petersburg State University

Commutators in Algebraic Groups

(based on joint work with Roozbeh Hazrat, Alexei Stepanov and Zuhong
Zhang)

In an abstract group, an element of the commutator subgroup is
not necessarily a commutator. However, the famous Ore conjecture,
recently completely settled by Ellers-Gordeev and by Liebeck-O'Brien-Shalev-Tiep,
asserts that any element of a finite simple group is a single commutator.

On the other hand, from the work of van der Kallen, Dennis and
Vaserstein it was known that nothing like that can possibly hold
in general, for commutators in classical groups over rings. Actually,
these groups do not even have bounded width with respect to commutators.

In the present talk, we report the amazing recent results which
assert that exactly the opposite holds: over any commutative ring
commutators have bounded width with respect to elementary generators,
which in the case of SL_n are the usual elementary transformations
of the undergraduate linear algebra course.

Technically, these results are based on a further development of
localisation methods proposed in the groundbreaking work by Quillen
and Suslin to solve Serre's conjecture, their expantion and refinement
proposed by Bak, localisation-completion, further enhancements implemented
by the authors (R.H., N.V, and Z.Zh.), and the terrific recent method
of universal localisation, devised by one of us (A.S.)

Apart from the above results on bounded width of commutators, and
their relative versions, these new methods have a whole range of
further applications, nilpotency of K_1, multiple commutator formulae
and the like, which enhance and generalise many important results
of classical algebraic K-theory.

In fact, our results are already new for the group SL_n, and time
permitting I would like to mention further related width problems
(unipotent factorisations, powers, etc.) and connections with geometry,
arithmetics, asymptotic group theory, etc.

30-minute
Talks

Bermudez, Hernando
Fields Institute, Emory University

Degree 3 Cohomological Invariants
of Split Quasi-Simple Groups

In this talk I will discuss the results of a recent joint work
with A. Ruozzi on degree 3 cohomological invariants of groups which
are neither simply connected nor adjoint. Using recent results of
A. Merkurjev we obtain a description of these invariants and we
show how our results relate to previous constructions. We also obtain
further applications to algebras with orthogonal involution.

MacDonald, Mark
Lancaster University

Reducing the structural group by using
stabilizers in general position

For a reductive linear algebraic group G (over the complex numbers),
all linear representations have the property that on an open dense
subset of V, the stabilizers are all conjugate to each other. This
is a result of Richardson and Luna. If H is an element of that conjugacy
class, then any G-torsor (over a field extension of the complex
numbers) is induced from an N_G(H)-torsor; in other words, we can
reduce the structural group from G to the normalizer of H. This
implies that the essential dimension of G is bounded above by that
of N_G(H). I will discuss how this extends to more general base
fields, in particular those of prime characteristic. The examples
of G=F_4 and G=E_7 will be considered.

Muthiah, Dinakar
Brown University

Some results on affine Mirkovic-Vilonen
theory

MV (Mirkovic-Vilonen) polytopes control the combinatorics of a
diverse array of constructions related to the representation theory
of semi-simple Lie algebras. They arise as the moment map images
of MV cycles in the affine Grassmannian. They describe the combinatorics
of the PBW construction of the canonical basis. And they control
the submodule behavior of modules for preprojective algebras and
KLR algebras. Recently, there has been much work toward extending
this picture to the case of affine Lie algebras. I will give a brief
overview of the current state of affairs, focusing on some rank-2
results (joint with P. Tingley) and some type A results on MV cycles.

Pollio, Timothy
University of Virginia

The Multinorm Principle

The multinorm principle is a local-global principle for products
of norm maps which generalizes the Hasse norm principle. Let L_1
and L_2 be finite separable extensions of a global field K. We say
that an element of the multiplicative group of K is a local multinorm
if it can be written as a product of norms of ideles from L_1 and
L_2 and we say that such an element is a global multinorm if it
can be written as a product of norms of field elements from L_1
and L_2. Then the pair of extensions L_1, L_2 satisfies the multinorm
principle if every local multinorm is a global multinorm. Two basic
problems are to determine which pairs of extensions satisfy the
multinorm principle and to describe the obstruction to the multinorm
principle which is defined as the group of local multinorms modulo
the group of global multinorms. I will discuss what is known about
each problem. In particular, I will sketch the computation of the
obstruction for pairs of abelian extensions using class field theory,
group cohomology, and the theory of Schur multipliers. I will also
outline a purely cohomological approach to the multinorm problem
which is based on the identification of the obstruction with the
Tate-Shafarevich group of the associated multinorm torus.

Rosso, Daniele
University of Chicago

Mirabolic Convolution Algebras

Several important algebras in representation theory, like Iwahori-Hecke
algebras of Weyl groups and Affine Hecke Algebras, can be realized
as convolution algebras on flag varieties. Some of these constructions
can be carried over to the 'mirabolic' setting to obtain other interesting
algebras. We will discuss the convolution algebra of GL(V)-invariant
functions on triples of two flags and a vector, which was first
described by Solomon, and its connections to the cyclotomic Hecke
algebras of Ariki and Koike.